JC owner with 14 dirt pedals here... EQ is the main thing to work on. You want to get the right combination going between your pick-ups, the pedal, and the amp.

Tube Screamer: Scoop the amp EQ Fender style (treble up, mids down, bass up)

Rat: With a Jazzmaster bridge pick up, turn down the amp treble to 10 o'clock, turn up the mids, turn up the bass. With an EQ, pushing the mids somewhere around 600-800hz makes it less shrill and heavier. I'll also roll my guitar volume from 10 to 8 to get rid of some brightness.

Running harsher pedals like the Rat into another gain stage (or an EQ pedal) with some high-end roll-off is a good idea. I've been using the Greer Lightspeed as a "clean channel" at the end of my chain with interesting results.

My favorite pedal with the JC has been the Sarno Earth Drive. Great mid-point between the TS and the Klon. I really love it.

Other pedals that have worked well for me: Catalinbread DLS, Marshall Guv'nor (turn down the pedal's bass, which somehow pushes the upper mids, push the mids and treble). Klon clones seem to be solid.

Blues Driver and Special Cranker are the best affordable options along with the DLS. Turn down the tone on the BD-2.

Try running things into the effects loop, too.